Phrygian Cap and 'Le Petit Chaperon Rouge' of Charles Perrault

Above: Buste of Attis as a child, wearing the Phrygian cap. Parian marble, 2nd century AD, probably during the reign of Emperor Hadrian.

The Phrygian cap or liberty cap was adopted as the symbol of liberty during the French revolution in 1790, but dates back to pre-Roman times. During the Roman era it the Phrygian Cap was worn as a symbol of a slave having achieved his freedom.

Above: Self Portrait with a Phrygian Cap by Anne-Louis Girodet de Roussy-Trioson - 1792

French author Charles Perrault (1628-1703) is reputed to have  created the first printed version of 'Le Petit Chaperon Rouge' in 1697, and therefore he is attributed with dressing the little girl in her first red hat or hood.
Here I would just like to ponder whether and why Charles Perrault introduced the little girls red hat to the story, which had previously been known as 'The False Grandmother' or the 'Story of the Grandmother' and is this hat a Phrygian Cap?

Above: Bust of Marianne (anonymous artist) Luxembourg Palace.

In France this tale is said to have been recounted by peasants since the 10th century.
It is however impossible to state definitively that Charles Perrault gave the girl a red Phrygian Cap.
But surely it is no coincidence that this is the peasants hat that within a hundred years was adopted as the symbol of the French Revolution and that it was depicted as worn by Marianne the Goddess of Liberty.

Did the peasant girl of the fairy tale already have this hat in one of the oral versions that Charles had heard? Or was this hat commonly worn by French peasant girls at this time?

"One serious difficulty in applying a historical  perspective to fairy tales in particular (as a subset of the larger category of folktale) is that the fairytale genre appears to be based more on fantasy than fact. (Legends are generally conceded to contain more historical reference than fairy tales, which are typically set in no one place or time.) Another obstacle to extrapolating history from fairy tales is that it is often impossible to ascertain with any degree of accuracy precisely when a specific tale first came into existence."  Hans-Wolf Jäger, Little Red Riding Hood edited by Alan Dundes, P89

It is impossible to tell, for as usual, it is very difficult to get a true view as to when and who was wearing this garment in the 17th century, especially in the sparsely documented peasant population, but it cannot have sprung from nowhere into the fore, as the garment of the revolution.

I also feel that it is not just coincidence that 'Little Red Riding Hood' as we understand her today, has imbued many of the strengths and virtues of Marianne and hence become an emblem of the fight for equality and justice.